**2. Integration methods for first- and second-order operators**

Computational fluid dynamics (CFD) is, in part, the art of replacing the governing partial differential equations of fluid flow with numbers and advancing these numbers in space and time to obtain a final numerical description of the flow [9]. To achieve this goal, accurate spatial and temporal discretization becomes necessary. CFD methods on unstructured grids mainly fall into three categories: finite element (FE) methods, finite volume (FV) methods, and DG methods.

#### **2.1 Finite volume methods**

Finite volume methods assume that the underlying solution is constant in each cell, and a Riemann problem [10] arises as discontinuity occurs at the boundary where two elements are adjacent to each other. For first-order operator

$$\nabla \cdot \mathbf{f}(u),\tag{1}$$

the finite volume method gives the formulation as

$$\int\_{\Omega\_{cl}} \nabla \cdot \mathbf{f}(u) d\Omega = -\int\_{\Gamma\_{cl}} \mathbf{n} \cdot \mathbf{f}(u) d\Gamma,\tag{2}$$

where Ω*el* is the volume of the element, and Γ*el* is the boundary faces of the element. This implies that only the normal fluxes through the element faces **n** � **f**ð Þ *u* appear in the discretization. For operators with second-order derivatives, the

*Hyperbolic Navier-Stokes with Reconstructed Discontinuous Galerkin Method DOI: http://dx.doi.org/10.5772/intechopen.109605*

integration is no longer obvious [11]. A number of strategies have been devised to circumvent this limitation. One of the popular methods is to evaluate the first-order derivatives in the first pass over the mesh and then obtain the second-order derivatives in a subsequent pass. A new hyperbolic Navier–Stokes (HNS) formulation is given in Ref. [12], where the gradients of density, velocity, and temperature are introduced as auxiliary variables. Therefore, the second-order derivatives can be integrated like any conservative variables in NS equations.

#### **2.2 Finite element methods**

Finite element methods are classically used for elliptic or parabolic problems. In the finite element method, a given domain is viewed as a collection of sub-domains, and over each sub-domain. Then, it becomes easier to represent a complicated function as a collection of simple polynomials [13]. As shown in **Figure 1**, the underlying solution is continuous at the element boundary. For second-order operator

$$\nabla \cdot (\nabla u), \tag{3}$$

FE formulation gives

$$\int\_{\Omega} w \nabla \cdot (\nabla u) d\Omega = \int\_{\Omega} \left(\nabla u\right) \cdot (\nabla w) d\Omega - \int\_{\Gamma} w \mathbf{n} \cdot (\nabla u) d\Gamma,\tag{4}$$

where *w* is a set of test functions. Volume integration is performed in the entire computation domain Ω, and boundary integration is performed at the boundary Γ of the entire domain. In **Figure 1**, Ω includes elements 1–6. Γ includes the left boundary of element 1 and the right boundary of element 6.

#### **2.3 Discontinuous Galerkin methods**

DG methods combine the advantages of finite volume and finite element methods. While assuming the underlying solution to be polynomial on each element, the Riemann problem is solved at the element boundary. For the first-order operator (Eq. (1)), the DG formulation gives

ð Ω*el w***∇** � **f**ð Þ *u d*Ω ¼ ð Ω*el* **f**ð Þ� *u* ð Þ **∇***w d*Ω � ð Γ*el w***n** � **f**ð Þ *u d*Γ, (5)

**Figure 1.**

*Piece-wise approximation of a function as u x*ð Þ¼ <sup>P</sup><sup>6</sup> *el*¼<sup>1</sup> *uel*ð Þ *<sup>x</sup> in finite element methods.*

In practice, *w* can be easily taken as the polynomial basis of the underlying DG solution. For second-order operator (Eq. (3)), DG formulation gives

$$\int\_{\Omega\_{d}} w \nabla \cdot (\nabla u) d\Omega = \int\_{\Omega\_{d}} (\nabla u) \cdot (\nabla w) d\Omega - \int\_{\Gamma\_{d}} w \mathbf{n} \cdot (\nabla u) d\Gamma. \tag{6}$$

As the DG method assumes that the solution is distributed as a polynomial function on each element, the value of **∇***u* is given explicitly on Ω*el*. However, **∇***u* becomes discontinuous at the element boundary Γ*el*. The flux can be estimated with BR1 method [14], BR2 method [15], or direct discontinuous Galerkin (DDG) method [16, 17]. One can achieve high-order accuracy while retaining the compactness of the stencil. Meanwhile, DG methods are especially suitable for hyperbolic systems, treatment of nonconforming meshes, and implementation of the hp-adaptive method. However, the DG method suffers from a number of weaknesses. In particular, how to reduce computing costs and how to discretize and efficiently solve elliptic/parabolic equations remain challenging issues in DG methods [7].

#### **2.4 Reconstructed discontinuous Galerkin methods**

In rDG methods, a higher-order solution is reconstructed based on the underlying solution. (Eq. (5)) becomes

$$\int\_{\Omega\_{d}} \mathbf{f}\left(\boldsymbol{u}^{R}\right) \cdot (\nabla \boldsymbol{w}) d\Omega - \int\_{\Gamma\_{d}} \boldsymbol{w} \mathbf{n} \cdot \mathbf{f}\left(\boldsymbol{u}^{R}\right) d\Gamma,\tag{7}$$

and (Eq. (6)) becomes

$$\int\_{\Omega\_{d}} \left(\nabla u^{R}\right) \cdot \left(\nabla w\right) d\Omega - \int\_{\Gamma\_{d}} w \mathbf{n} \cdot \left(\nabla u^{R}\right) d\Gamma,\tag{8}$$

where *uR* is a higher-order solution reconstructed from the underlying solution *u*.
